Monotonic solutions of a class of quadratic integral equations of Volterra type

Using the technique associated with measures of noncompactness we prove the existence of monotonic solutions of a class of quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval.     

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space $X$. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator $A$. Finally we give an example to illustrate the applications of the abstract results.     

A new type of matrix quadratic equation

We derive a number of properties of a new type of matrix quadratic equation, including the existence of a maximal solution.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Positive solutions to a nonlinear abel-type integral equation on the whole line

We consider the existence of positive solutions to the nonlinear integral equation

where g is a continuous, nondecreasing function such that g(0) = 0. We show that the equation always has nontrivial solutions and we give a necessary and sufficient condition for the existence of solutions u such that u(x) > − ∞. We also provide a condition which ensures that all the nontrivial solutions experience the blow-up behaviour.     

Monotonic solutions of Urysohn integralequation on unbounded interval

In this paper, we investigate a nonlinear Urysohn integral equation on unboundedinterval. We show that under some assumptions that the equation has monotonic solutions belonging to the space of functions being Lebesgue integrable on unbounded interval. The main tool used in our study is the technique associated with measures of weak noncompactness and measures of noncompactness in strong sense.     

How to solve a quadratic equation?

One of the author's favorite experiences in high school mathematics was learning how to solve quadratic equations. As is usual with life, things are not as simple as they were in high school. There are several problems with the quadratic formula. In this article, the author talks about these problems and various solutions to them.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation

In this paper, by using the integral bifurcation method, we study a nonlinear dispersive equation. Some new soliton-like solutions and some compacton-like periodic wave solutions are obtained. Their dynamic characters are investigated and the profiles are given by the mathematical software Maple. From the graphs of some soliton-like solutions, we find that their profiles are transformable.     

System analysis via integral quadratic constraints

This paper introduces a unified approach to robustness analysis with respect to nonlinearities, time variations, and uncertain parameters. From an original idea by Yakubovich (1967), the approach has been developed under a combination of influences from the Western and Russian traditions of control theory. It is shown how a complex system can be described, using integral quadratic constraints (IQC) for its elementary components. A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality. A systematic computational approach is described, and relations to other methods of stability analysis are discussed. Last, but not least, the paper contains a summarizing list of IQCs for important types of system components     

On a weakly singular Volterra integral equation

The need for providing reliable numerical methods for the solution of weakly singular Volterra integral equations ofI ^st Kind stems from the fact that they are connected to important problems in the theory and applications of stochastic processes. In the first section are briefly sketched the above problems and some peculiarities of such equations. Section 2 described the method for obtaining an approximate solution whose properties are described in section 3: such properties guarantee that our approximate solution always oscillates around the rigorous one. Section 4 discusses the applicability to our case of some classical bounds on the errors. The remaining sections are all devoted to the construction of upper bounds on the oscillating error in order to reach a high degree of reliability for our solution. All the bounds are independent on the numerical method which is employed for obtaining the numerical solution. In section 5 is derived a Volterra II Kind integral equation by subtracting to the original kernel the weak singularity, while in section 6 is given an upper bound to the error in the case of Wiener and Ornstein-Ühlenbeck kernels with constant barriers. Such a bound is generalized to other kinds of barriers in section 7 while in section 8 is suggested an approximation of the Kernel for the O. Ü. case with constant barriers and by means of it is given an explicit bound for the error in terms of Abel's transform of the known term in the original integral equation. A rough estimation of the error is also given under the assumption that \(y(t) - \int\limits_0^t {K(t,\tau )\tilde x(\tau )d_\tau [\tilde x(\tau )} \) denotes any approximate solution of (1a) obtained by any method] can be approximated by means of a sinusoidal function. In section 9 is derived another kind of bound, for constant barriers, by using the approximate Kernel of section 7 and classical results.     

How to solve a quadratic equation. Part 2

We derived an algorithm to find the real roots of the homogeneous quadratic equation, Ax/sup 2/+2Bxw+Cw/sup 2/=0. Because the equation is homogeneous, a root consists of an [x, w] pair where any nonzero multiple represents the same root. We strove to find an algorithm that didn't blow up no matter what values of A, B, and C we were given, including various combinations of zeroes. At the end of the article the author wrote the final algorithm in tabular form.     

Regularization of a Volterra integral equation by linear inequalities

The determination of a monotone nonincreasing and convex response function arising in reservoir mechanics is investigated from the computational point of view. Regularization by linear inequalities yields the means for overcoming the ill-posedness of the considered convolution type integral equation. In order to find efficient numerical solutions and adapted approach for solving the associated constrained least squares problems is developed. Some simulation studies complete the paper.     

Stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic Riccati equation

This note addresses the jump linear quadratic problem of Markov jump linear systems and the associated algebraic Riccati equation. Necessary and sufficient conditions for stability of the optimal control and positiveness of Riccati solutions are developed. We show that the concept of weak detectability is not only a sufficient condition for the finiteness of cost functional to imply stability of the associated trajectory, but also a necessary one. This, together with a characterization developed here for the kernel of the Riccati solution, allows us to show that the control solution stabilizes the system if and only if the system is weakly detectable, and that the Riccati solution is positive-definite if and only if the system is weakly observable. The connection between the algebraic Riccati equation and the control problem is made, as far as the minimal positive-semidefinite solution for the algebraic Riccati equation is identified with the optimal solution of the linear quadratic problem. Illustrative numerical examples and comparisons are included.     

Finite-dimensional solutions of a modified Zakai equation

In risk-sensitive control a modified Zakai equation arises which includes an extra term related to the exponential running cost. We show that a wide class of finite-dimensional solutions related to the Beneš filter is possible as long as nonlinearities in the drift are canceled in an appropriate way by terms in the running cost. The support of NSERC Grant A7964 is gratefully acknowledged.